How to divide a face into a square grid?
Knife project  could help you:

subdivide a plane that will become your "shape cut" grid.



then over it (higer in Z) put you shape (just the outer edges, delete everything else)





then select 
- first the cutting shape
- then (shift-select) also the background subdivided plane, then search and use "knife project"



then, your plane now has a cut shaped exactly as you want



just invert the selection,



and delete everyhing else



now of course you can extrude this mesh